Microorganisms swimming through viscous fluids imprint their propulsion mechanisms in the flow fields they generate. Extreme confinement of these swimmers between rigid boundaries often arises in natural and technological contexts, yet measurements of their mechanics in this regime are absent. Here, we show that strongly confining the microalga Chlamydomonas between two parallel plates not only inhibits its motility through contact friction with the walls but also leads, for purely mechanical reasons, to inversion of the surrounding vortex flows. Insights from the experiment lead to a simplified theoretical description of flow fields based on a quasi-2D Brinkman approximation to the Stokes equation rather than the usual method of images. We argue that this vortex flow inversion provides the advantage of enhanced fluid mixing despite higher friction. Overall, our results offer a comprehensive framework for analyzing the collective flows of strongly confined swimmers.